Finite Group Databasebeta

← #385206

Subgroups — #385206

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1948C212yesmaximal, complemented
2C2×#1850C211yesmaximal, characteristic
3C2×#1807C212yesmaximal
4#13876C212yesmaximal
5#11323C212yesmaximal, complemented
6#11231C212yesmaximal, complemented
7#11095C211yesmaximal, characteristic
8#11095C211yesmaximal, characteristic
9#10908C212yesmaximal, complemented
10C24×C8C2211yescharacteristic
11C22×#106C2212yes—
12C22×#106C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C2×#388—22no—
18C2×#384C2212yes—
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1948—28no—
24#1850C418yescomplemented
25#1807—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#336C2212yes—
33#1442—22no—
34#1439—22no—
35#1436—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C24×C4C2311yescharacteristic
39C23×C8—24no—
40C23×C8—24no—
41C23×C8—41no—
42C23×C8—41no—
43C23×C8C2×C414yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4D411yescharacteristic
56C22×C22⋊C4D411yescharacteristic
57C22×C42—22no—
58C22×C42C2312yes—
59C2×#106—24no—
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—41no—
65C2×#106—42no—
66C2×#106—42no—
67C2×#106—42no—
68C2×#106—41no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8C2×C418yes—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74C2×C22⋊C8C2×C414yes—
75#388—48no—
76#384—24no—
77#384—42no—
78#380—24no—
79#380—24no—
80#380—42no—
81#380—42no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—42no—
84C4×C22⋊C4C2×C414yes—
85C4×C22⋊C4Q814yes—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95D412yes—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—41no—
98#340—41no—
99#340D418yes—
100#336C2×C418yes—
101C25C2×D411yescharacteristic
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4C22×C412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×Q812yes—
119C23×C4C2411yescharacteristic
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—216no—
133C22×C8—44no—
134C22×C8—44no—
135C22×C8—48no—
136C22×C8—44no—
137C22×C8C22×C412yes—
138C22×C8C22×C412yes—
139C22×C8C22×C414yes—
140C22×C8C2×D414yes—
141C22×C8#4114yes—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×D414yes—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—82no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4C4⋊C414yes—
180C2×C22⋊C4C4⋊C414yes—
181C2×C22⋊C4SD1618yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—44no—
187C2×C42—44no—
188C2×C42—82no—
189C2×C42C22×C414yes—
190C2×C42C2×Q814yes—
191#106—44no—
192#106—48no—
193#106—48no—
194#106—44no—
195C4⋊C8—216no—
196C22⋊C8—44no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—82no—
204#95C4⋊C418yes—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24C2×C4⋊C412yes—
213C24C2×SD1614yes—
214C24C22×D411yescharacteristic
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4C2×C4⋊C412yes—
276C22×C4C2×C4⋊C414yes—
277C22×C4C2×C4⋊C414yes—
278C22×C4C2×SD1614yes—
279C22×C4C2×SD1614yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D412yes—
282C22×C4C22×Q811yescharacteristic
283C22×C4C2×#4112yes—
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)11yescharacteristic
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+411yescharacteristic
288C22×C42−1+411yescharacteristic
289C22×C42−1+412yes—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—416no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8C4×D418yes—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—88no—
325C4⋊C4—88no—
326C22⋊C4—44no—
327C22⋊C4—88no—
328C22⋊C4—88no—
329C22⋊C4#106116yescomplemented
330C42—24no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—88no—
335C23—24no—
336C23—24no—
337C23—22no—
338C23—24no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—24no—
344C23—24no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23C2×#10614yes—
356C23C22×C4⋊C411yescharacteristic
357C23#53711yescharacteristic
358C23#53711yescharacteristic
359C23D4212yes—
360C23#54712yes—
361C23C22×SD1612yes—
362C23#57512yes—
363C23#57712yes—
364C23#57712yes—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4C2×#10618yes—
418C2×C4C2×C4×D412yes—
419C2×C4#51811yescharacteristic
420C2×C4#51911yescharacteristic
421C2×C4#52012yes—
422C2×C4#53612yes—
423C2×C4#53712yes—
424C2×C4#54112yes—
425C2×C4#54212yes—
426C2×C4#54411yescharacteristic
427C2×C4#54411yescharacteristic
428C2×C4D4211yescharacteristic
429C2×C4#54612yes—
430C2×C4D4×Q812yes—
431C2×C4#55111yescharacteristic
432C2×C4#55412yes—
433C2×C4#56011yescharacteristic
434C2×C4#56212yes—
435C2×C4#56311yescharacteristic
436C8—28no—
437C8—42no—
438C8—42no—
439C22—28no—
440C22—24no—
441C22—42no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22#231611yescharacteristic
449C22D4×C4⋊C412yes—
450C22#267212yes—
451C22#267511yescharacteristic
452C22#272211yescharacteristic
453C22C22×#10611yescharacteristic
454C22#291111yescharacteristic
455C22#291212yes—
456C22#324612yes—
457C22#324612yes—
458C22#326012yes—
459C22#326311yescharacteristic
460C22#326311yescharacteristic
461C22#326712yes—
462C22#327112yes—
463C22D4×SD1612yes—
464C22D4×SD1612yes—
465C22#333512yes—
466C22#333512yes—
467C22#334812yes—
468C22#335012yes—
469C4—28no—
470C4—42no—
471C4—44no—
472C4—84no—
473C4—84no—
474C4—162no—
475C4—162no—
476C4#232312yes—
477C4#235811yescharacteristic
478C4#237611yescharacteristic
479C4#267711yescharacteristic
480C4#270412yes—
481C4#274211yescharacteristic
482C2—44no—
483C2#1440211yescharacteristic
484C2#2195911yescharacteristic
485C2#2196711yescharacteristic
486C2D4×#10612yes—
487C2#2266212yes—
488C2#3066412yes—
489C2#3066912yes—
490C2#3084812yes—
491C2#3240812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#1948C2semidirect
5. #11323C2semidirect
6. #11231C2semidirect
9. #10908C2semidirect
24. #1850C4semidirect
329. C22⋊C4#106semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#19482
C2×#18502
C2×#18072
#138762
#113232
#112312
#110952
#109082